complementary subspace

Direct sum decomposition.

Let U be a vector spaceMathworldPlanetmath, and V,WU subspacesPlanetmathPlanetmath. We say that V and W span U, and write


if every uU can be expressed as a sum


for some vV and wW.

If in additionPlanetmathPlanetmath, such a decomposition is unique for all uU, or equivalently if


then we say that V and W form a direct sumMathworldPlanetmath decomposition of U and write


In such circumstances, we also say that V and W are complementary subspaces, and also say that W is an algebraic complement of V.

Here is useful characterization of complementary subspaces if U is finite-dimensional.

Proposition 1

Let U,V,W be as above, and suppose that U is finite-dimensional. The subspaces V and W are complementary if and only if for every basis v1,,vm of V and every basis w1,,wn of W, the combined list


is a basis of U.


  • Since every linearly independent subset of a vector space can be extended to a basis, every subspace has a complementPlanetmathPlanetmath, and the complement is necessarily unique.

  • Also, direct sum decompositions of a vector space U are in a one-to correspondence fashion with projections on U.

Orthogonal decomposition.

Specializing somewhat, suppose that the ground field 𝕂 is either the real or complex numbers, and that U is either an inner product spaceMathworldPlanetmath or a unitary space, i.e. U comes equipped with a positive-definite inner productMathworldPlanetmath


In such circumstances, for every subspace VU we define the orthogonal complementMathworldPlanetmath of V, denoted by V to be the subspace

V={uU:v,u=0, for all vV}.
Proposition 2

Suppose that U is finite-dimensional and VU a subspace. Then, V and its orthogonal complement V determine a direct sum decomposition of U.

Note: the PropositionPlanetmathPlanetmath is false if either the finite-dimensionality or the positive-definiteness assumptionsPlanetmathPlanetmath are violated.

Title complementary subspace
Canonical name ComplementarySubspace
Date of creation 2013-03-22 12:52:16
Last modified on 2013-03-22 12:52:16
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 11
Author rmilson (146)
Entry type Definition
Classification msc 15A03
Synonym algebraic complement
Defines complementary
Defines direct sum
Defines decomposition
Defines orthogonal complement